Moving Average (MA), Weighted MA, and Exponential MA

Moving averages are favored tools of active traders to measure momentum. The primary difference between a simple moving average, weighted moving average, and the exponential moving average is the formula used to create the average.

Simple Moving Average

The simple moving average (SMA) was prevalent before the emergence of computers because it is easy to calculate. Today’s processing power has made other types of moving averages and technical indicators easier to measure. A moving average is calculated from the average closing prices for a specified period. A moving average typically uses daily closing prices, but it can also be calculated for other timeframes. Other price data such as the opening price or the median price can also be used. At the end of the new price period, that data is added to the calculation while the oldest price data in the series is eliminated.

For a simple moving average, the formula is the sum of the data points over a given period divided by the number of periods. For example, the closing prices of Apple Inc (AAPL) from June 20 to 26, 2014, were as follows:

A five-period moving average, based on the prices above, would be calculated using the following formula:


$begin{aligned} &text{MA} = frac{ P_1 + P_2 + P_3 + P_4 + P_5 }{ 5 } \ &textbf{where:} \ &P_n = text{Price for time period} \ end{aligned}$​MA=5P1​+P2​+P3​+P4​+P5​​where:Pn​=Price for time period​

or:


$begin{aligned} &frac{ 90.90 + 90.36 + 90.28 + 90.83 + 90.91 }{ 5 } = 90.656 \ end{aligned}$​590.90+90.36+90.28+90.83+90.91​=90.656​

The equation above shows that the average price over the period listed was $90.66. Using moving averages is an effective method for eliminating strong price fluctuations. The key limitation is that data points from older data are not weighted any differently than data points near the beginning of the data set. This is where weighted moving averages come into play.

Weighted Moving Average

Weighted moving averages assign a heavier weighting to more current data points since they are more relevant than data points in the distant past. The sum of the weighting should add up to 1 (or 100 percent). In the case of the simple moving average, the weightings are equally distributed, which is why they are not shown in the table above.

For example:

The weighted average is calculated by multiplying the given price by its associated weighting and totaling the values. The formula for the WMA is as follows:


$begin{aligned} &text{WMA} = frac{ text{Price}_1 times n + text{Price}_2 times ( n – 1 ) + cdots text{ Price}_n }{ frac{ n times ( n + 1 ) }{ 2} } \ &textbf{where:} \ &n = text{Time period} \ end{aligned}$​WMA=2n×(n+1)​Price1​×n+Price2​×(n−1)+⋯ Pricen​​where:n=Time period​

The denominator of the WMA is the sum of the number of price periods as a triangular number. In the example from the table above, the weighted five-day moving average would be $90.62:


$begin{aligned} ( 90.90 times tfrac{ 5 }{ 15 } ) &+ ( 90.36 times tfrac{ 4 }{ 15 } ) + ( 90.28 times tfrac{ 3 }{ 15 } ) \ &+ ( 90.83 times tfrac{ 2 }{ 15 } ) + ( 90.91 times tfrac{ 1 }{ 15 } ) = $90.62 \ end{aligned}$(90.90×155​) ​+ (90.36×154​) + (90.28×153​)+(90.83×152​) + (90.91×151​)=$90.62​

In this example, the recent data point was given the highest weighting out of an arbitrary 15 points. You can weigh the values out of any value you see fit. The lower value from the weighted average above relative to the simple average suggests that recent selling pressure could be more significant than some traders anticipate. For most traders, the most popular choice when using weighted moving averages is to use a higher weighting for recent values. (For more information, see: Moving Average Tutorial.)

Exponential Moving Averages

Exponential moving averages (EMAs) are also weighted toward the most recent prices, but the rate of decrease between one price and its preceding price is not consistent. The difference in the decrease is exponential. Rather than every preceding weight being 1.0 smaller than the weight in front of it, there might be a difference between the first two period weights of 1.0, a difference of 1.2 for the two periods after those periods, and so on. The formula for EMA is


$begin{aligned} &text{EMA} = text{Price}_t times k + text{SMA}_y times ( 1 – k ) \ &textbf{where:} \ &t = text{Today} \ &k = frac { 2 }{ text{Number of days in period} + 1 } \ &text{SMA} = text{Simple Moving Average of closing price} \ &text{for the number of days in the period} \ &y = text{Yesterday} \ end{aligned}$​EMA=Pricet​×k+SMAy​×(1−k)where:t=Todayk=Number of days in period+12​SMA=Simple Moving Average of closing pricefor the number of days in the periody=Yesterday​

Calculating an EMA involves three steps. The first step is to determine the SMA for the period, which is the first data point in the EMA formula. Then, a multiplier is calculated by taking 2 divided by the number of periods plus 1. The final step is to take the closing price minus the prior day EMA times the multiplier plus the prior day EMA. (For related reading, see: How is the Exponential Moving Average (EMA) Formula Calculated?)

Which Moving Average is More Effective?

Because an exponential moving average (EMA) uses an exponentially weighted multiplier to give more weight to recent prices, some believe it is a better indicator of a trend compared to a WMA or SMA. Some believe that the EMA is more responsive to changes in trends. On the other hand, the more basic smoothing provided by the SMA may render it more effective for finding simple support and resistance areas on a chart. In general, moving averages smooth price data that can otherwise be visually noisy.

The functions of an EMA and a WMA are similar, they rely more heavily on the most recent prices and placing less value on older prices. Traders use these EMAs and WMAs over SMAs if they are concerned that the effects of lags in data may reduce the responsiveness of the moving average indicator.

All moving averages have a significant drawback in that they are lagging indicators. Since moving averages are based on prior data, they suffer a time lag before they reflect a change in trend. A stock price may move sharply before a moving average can show a trend change. A shorter moving average suffers from less lag than a longer moving average.

Still, this lag is useful for certain technical indicators known as moving average crossovers. The technical indicator known as the death cross occurs when the 50-day SMA crosses below the 200-day SMA, and it is considered a bearish signal. An opposite indicator, known as the golden cross, is created when the 50-day SMA crosses above the 200-day SMA, and it is considered a bullish signal. (For related reading, see: How to Use a Moving Average to Buy Stocks.)